Helen Czerski does temperature on BBC4

Dr Helen Czerski is doing a series on tmeperature on BBC4 at the moment. I enjoyed the first programme. The link will be valid for a month or so http://www.bbc.co.uk/programmes/b09rzq05

Joseph Black

I hadn't come across Joseph Black before I stood in front of his portrait, but I've been teaching his ideas for years. It turns out he was the person who came up with latent heat and found that different materials have different specific heats. It started with the discovery that if you heat ice at its melting point, its temperature does not go up until all of the ice has melted and become water.

Aeroplanes over High Pike again

Ah! The joy of going up High Pike again in the company of a smart phone owner. When first spotted the Airbus was at 31000 feet and over Ullswater.

By the time I took this photo it was down to 29000 feet.and half way between us and the motorway. 29000 is 9.4km. Half way to the motorway is about 10km. By Pythagoras, that about 14km away. The plane is about 70m long. Angle subtended is 70/14000 = 0.005 radians. That's 0.3 of a degree. If my finger is worth 1 degree, then the plane is worth ... well, I've blown up the image and it is 2/12 = 0.2 of a degree. Not too bad.

Cross Fell: Singing gatepost at Kirklands

The wind was blowing across the hole in the gate post on the way up Cross Fell. It was oscillating between a low note that we thought was around middle C and a much higher note. So which type of stationary wave was being formed. Both ends of the top are closed so let's try the idea that there is a node at each end like on a guitar string. The wave is then a single loop and the wavelength is twice the length of the tube. The tube was about 1 metre long so wavelength = 2m. Speed of sound in air is approximately 340 metres per second. So frequency = wave speed/wavelength = 340/2 = 170Hz. The other alternative is a node at one end and an antinode at the hole which is a quarter of a wavelength and might be worth 70cm. So wavelength = 2.8 metres and frequency = 121 Hz. Middle C is 261 Hz. I'd say first mode is most likely. The higher note would be the second harmonic at 340Hz.

Identity matrix in Dirac notation

I have been working through Leonard Susskind's book Qunatum Mechanics: The Theoretical Minimum. I have decided that I like Dirac notation. I have always found matrices easy to understand so I have been translating Dirac notation into my language. I decided to work in the ordinary 3D Cartesian space. This illustrates what is going on although it is not proof that it works in any coordinate system (basis). So here is the x-axis unit vector. I've called it i. Dirac writes it as |i>

 But there is also a line vector version. This is called the DUAL. Dirac writes it as . Dirac called this bracket notation because of the outside brackets, or bra-ket notation. The dual is called the ket. The inner product gives just a number.

 I then multiplied them the other way round and found out that I got a matrix. This is called taking the OUTER PRODUCT. In quantum mechanics, a matrix is called an OPERATOR. I did it with all 3 basis vectors for x, y and z coordinates.

 When you add the three matrices together you get the identity matrix I. It is like the number 1 for matrices. If you multiply any matrix by I, you get the same matrix back. I did it for the limited case of 3D Cartesian space but here is the general result for any orthogonal basis. This is what I was trying to understand. More on orthogonal later but in 3D space it means the vectors are at 90 degrees to each other.

Lord Kelvin

I met Lord Kelvin in Edinburgh. I was surprised to find later that he was yet another Cambridge graduate! I knew about his work on the Laws of Thermodynamics but I hadn't realised that he had been heavily involved in the project to lay the transatlantic cable to America.

Penrose Tiling Phone Case

I was shown this wonderful phone case by Will. It's a Penrose Tiling. I confess to having never heard of them but here's the explanation https://en.wikipedia.org/wiki/Penrose_tiling I think they will take some investigation and might make a good patchwork project. The case is an example of P2 tiling. My new ambition is to see the floor in the Maths Institute at Oxford.

Wigton Drip: an update

I went back past the drip last night and was sure that it was dripping faster than the once every two seconds that I calculated last week. So I counted the drips in 5 seconds and got 12. That's a frequency of more than 2 Hz. So were my calculations wrong or does the frequency depend on the conditions? Will have to wait until it rains again...

Analemmatic sundial in Williamson Park, Lancaster

We found another one of the DIY sundials in Lancaster. It is anlemmatic because the shadow maker (the gnomon) is vertical and not angled to the latitude. Vertical because it is a person. The upshot is that to get the shadow to fall correctly you need to stand in a different place each month. Here are the settings:

There were three things of interest:
1. It was set for British Summer Time. What a brilliant idea! When are the shadows likely to be best..
2. It has corrections given each month in minutes. The light curve was at the top but was very small.
3. An anlemma https://en.wikipedia.org/wiki/Analemma is a figure of eight and yet the monthly track doesn't seem to represent that.

Bridge over the Thames at Abingdon: F = -gradV

We were sat in Annie's cafe by the bridge in Abingdon and looked up our height above sea level. It is 120 metres above sea level. This difference is enough to drive all that water about 100 miles down to the sea. I had just been reading that the force acting on something = - potential energy gradient. That can be written as -grad V where grad is a differential operator. Potential energy is a scalar quantity with each point on the landscape having its own value. But the numbers vary from place to place and thus we can say that there are places where the potential energy is going up and places where it is going down if we move across the landscape. Hence the gradient. In one dimension it is the simple differential dV/dx.

Let's say that we have to go distance x down to the sea. There is potential energy mgh where we are and 0 at the sea by definition. The gradient thus becomes mgh/x. As shown below, the component of the gravitational force mg driving the motion is mgsin(theta). Now it works if sin(theta) = h/x, which is true if the distance x is the distance measured down the slope to the sea. It doesn't matter too much over such a large distance because the difference in length between the two long sides will be virtually impossible to measure.

Oxford gramophone at Westgate

Stood at the bus stop opposite the Westgate Centre, I spotted the gramophone in the graphic for Oxford in the window opposite. I realised that I have never thought about how a gramophone works. They were wind up and so not electrical. I have taught about record players but their needles oscillated magnets between coils to produce an electrical signal that was then amplified. I have looked up gramophones and I had guessed correctly. The needle is made to oscillate backwards and forwards by the groove on the recording. The oscillating needle is attached to a diaphragm which makes the air oscillate. The horn is designed to make this as loud as possible.

Losing to a Fermi question

Our team came equal first in a fiendish pub quiz. The tie breaker was what I'd call a Fermi question. I have used these a lot with classes so I was ashamed to lose. It was late, we were only given a couple of minutes to think, we'd traveled all day and I was tired - those are my excuses. This was the question: how many Kit-Kat fingers are consumed globally in one minute? I started with the world population at about 6000 million and worked downwards. We stopped at 500,000. The winners guessed 100,000. Both were wildly out as the answer given was about 400. Later Mrs B came up with a better way of thinking about it: How many bars can roll off the end of the production line in a minute. Enrico Fermi was the great Italian physicist. When he taught in Chicago he pioneered these seemingly random questions as a way of getting students to model the way that physicists develop models and calculations. https://en.wikipedia.org/wiki/Fermi_problem I can even use ones for homework where the answer is not available on the Internet. My favourite of these is "how many take-away pizzas were sold in Cumbria last year?"

Insulators and Earth at Sandwell Valley

 We were at Forge Mill Farm in Birmingham and I noticed that the insulators on these seemingly identical pylons had different colours. I can't find any reason. 11 insulators might be 132 kV wire (see http://www.epemag.net/electricity-generation-pylons.html which says 9 are needed) As I had binoculars with me I inspected the thin wire that comes down the left hand side of the pylon. I'm assuming it is an earth wire. I read https://electronics.stackexchange.com/questions/73726/what-is-the-purpose-of-the-thin-wire-at-the-top-of-power-transmission-lines. I'm wondering if it is actually because the guard wire along the top is used for communication in this case. This needs more work but I started at https://en.wikipedia.org/wiki/Optical_ground_wire

Wigton Drip 2

I spotted a regular drip from a gutter that was producing concentric ripples in the puddle below.

I put my foot into the frame for scale and used the multi-shot function on my camera. The wavelength is about the width of my foot so let's say 10cm.

These are two consecutive frames. It's hard to tell but I think the wavefronts have advanced about 1cm.

To find the time between two consecutive frames, I photographed a watch.

The first frame is a above and the sixth frame is below. It took 5 frames to go 1 second so let's say the time for 1 frame is 0.2 seconds.

If the wavefront advances 1cm in 0.2 seconds, the wave speed is 5 cm per second. Now frequency = wave speed / wavelength = 5/10 = 0.5 Hz or one drip every 2 seconds.

Wigton Drip 1

We found this ice stalagmite under a drip from the greenhouse gutter. Why did the water freeze and build upwards instead of just flowing out horizontally? It must have fallen as a liquid. It occurred to me that ice must be a reasonable conductor of thermal energy because it feels cold to the touch. Its thermal conductivity is about 2 Watts per metre per Kelvin, compared to 0.59 for water and 7.8 for manganese, which is apparently the lowest for a metal. The figures suggest that the ice in the bucket can conduct thermal energy away from the falling drip fast enough to cool it below freezing before the liquid drop has time to run off.

Glen Clova defibrillator

Defibrillator units like this are becoming a common sight. Often they are inside the phone box but there is no mobile signal at Glen Clova so the phone is still in use. A defibrillator is basically a large capacitor with a small time constant that means it can discharge a lot of energy through the heart in a short space of time. The idea is to knock the heart muscle back into its regular rhythm. This site has excellent details about the wiring: http://www.frca.co.uk/article.aspx?articleid=100392 I was interested to see that there is also an inductance in the discharge circuit. It says that it is there to prolong the duration of the current flow. I would have thought that you just go for a bigger resistor to increase the time constant because time constant = capacitance x resistance but maybe the bigger resistance increases the time by decreasing the current. Using an inductance probably gives longer time for the same higher current.

Vector calculus: Experimenting with divergence

 I have never really understood what divergence does in a vector field although I was able to pass exams because I learned to calculate it. It occurred to me that if I made my own field in 2-dimensions then I could test my own hypothesis about what divergence means. I made the vector field F above. It is a vector field because there is an arrow at each coordinate showing the size and direction of the function F. I have set it up so that the function has an x component Fx along the i direction that is equal to the size of the x coordinate but there is no component in the j direction. The word divergence in the English language means that things that were travelling parallel to each other split apart and separate due to direction change. That's how I pictured divergence in a vector field. This vector field has been set up because the vectors change size but remain parallel. My idea would mean no divergence. So here comes the calculation.

The upside down triangle is the differential operator nabla or del. Here it is differentiating by the dot product so it is to do with the gradient along the line of the vector; perhaps the rate of change of size along the direction of the vector. The divergence calculates out as 1. So there is divergence of this vector field even though the vectors remain parallel. So the stretching of the vector, the increase in size as it goes along, constitutes divergence.